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Applied Math Seminar
Homogenization and inverse homogenization of divergence form elliptic
operators with a continuum of scales.
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We consider divergence form elliptic operators with bounded ($L^\infty$) coefficients. Although solutions of these operators are only Hölder-continuous, we show that they are differentiable ($C^{1,\alpha}$) with respect to harmonic coordinates. We deduce that numerical homogenization can be extended to situations where the medium is characterized by a continuum of scales. Next we consider the inverse homogenization of these operators which is known to be a non-linear ill posed problem. We show how this problem can be transformed into the search of an optimal solution within a linear space by putting scalar conductivities into one to one correspondence with curvatures of concave functions. We propose a new class of robust and converging algorithms for electrical impedance tomography based on these results. (Parts of this talk are joint works with Lei Zhang, Roger Donaldson, Mathieu Desbrun and Yiying Tong) |